by Bavi_H, 3/25/2008

When my sister hung my mirror too low, my head was cut off in the reflection. I made up the following problem in order to find the best height to hang a mirror, but wasn't able to solve it until recently.

How high should you hang a mirror so that the angles of the left over space above and below your reflection in the mirror are equal?

`E` is the height of your eyes.

`C` is the distance from your eyes to the top of your head. (*C* for *crown*.)

`L` is the length of the mirror.

`D` is the distance from you to the wall.

`h` is how high from the floor you should hang the mirror.

`θ` is the angle of the left over space above and below your reflection.

I was able to write an equation setting the angle of the left over space above the reflection equal to the angle of the left over space below the reflection:

But I didn't know how to proceed to solve for `h`. With some help from mathematicians at the Wikipedia Reference Desk and the automated solver at Quickmath I've finally found the solution.

Here's an example solution using values for my mirror and body measurements.

Using the same values for `E`, `C`, and `L`, here are graphs that show how `h` and `θ` change as `D` increases.

When I actually tested the results by positioning my mirror and looking at my reflection, I noticed that my feet were taking up room that I hadn't accounted for. Here's an updated solution.

How high should you hang a mirror so that the angles of the left over space above and below your reflection in the mirror are equal?

`E` is the height of your eyes.

`C` is the distance from your eyes to the top of your head. (*C* for *crown*.)

`F` is the distance your feet stick out from your eyes.

`L` is the length of the mirror.

`D` is the distance from you to the wall.

`h` is how high from the floor you should hang the mirror.

`θ` is the angle of the left over space above and below your reflection.

I used Quickmath again to solve the equation for `h`.

Here's an example solution using values for my mirror and body measurements.

Using the same values for `E`, `C`, `F`, and `L`, here are graphs that show how `h` and `θ` change as `D` increases. The previous model is shown in gray, the updated model is shown in red.

Wikipedia Reference Desk

http://en.wikipedia.org/wiki/Wikipedia:Reference_desk

Volunteers answer reference desk-type questions.

Solving equation with inverse tangents

http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2008_March_25#Solving_equation_with_inverse_tangents

My post to the Wikipedia Mathematics Reference Desk where I got advice about how to solve the equation of this problem.

Quickmath

http://www.quickmath.com/

Automated solvers for equations, matrices, and other math problems.